Zero-error communication under discrete-time Markovian dynamics

2024-02-28

AI Summary

This study addresses the problem of achieving zero-error, long-term information storage and perfect recovery in open quantum systems governed by discrete-time Markovian dynamics, where environmental coupling typically causes irreversible information loss. Methodologically, we establish, for the first time, precise equivalences between mixing (or asymptotic entanglement-breakingness) and the impossibility of zero-error classical/quantum communication. We introduce a universal encoding principle based on the peripheral space and derive an explicit formula for the zero-error information capacity. Leveraging operator-algebraic techniques and analysis of the peripheral spectrum, we obtain a tight upper bound on the recovery time beyond which information becomes irretrievable. Furthermore, in the non-mixing and non-asymptotically-EB regimes, we provide exact analytical expressions for the amount of protectable information.

Technology Category

• Machine Learning: Information Theory
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Consider an open quantum system with (discrete-time) Markovian dynamics. Our task is to store information in the system in such a way that it can be retrieved perfectly, even after the system is left to evolve for an arbitrarily long time. We show that this is impossible for classical (resp. quantum) information precisely when the dynamics is mixing (resp. asymptotically entanglement breaking). Furthermore, we provide tight universal upper bounds on the minimum time after which any such dynamics'scrambles'the encoded information beyond the point of perfect retrieval. On the other hand, for dynamics that are not of this kind, we show that information must be encoded inside the peripheral space associated with the dynamics in order for it to be perfectly recoverable at any time in the future. This allows us to derive explicit formulas for the maximum amount of information that can be protected from noise in terms of the structure of the peripheral space of the dynamics.

Problem

• Studying zero-error information storage in quantum systems with Markovian dynamics
• Determining when classical or quantum information becomes irretrievable due to dynamics
• Deriving maximum protectable information limits using peripheral space structure

Innovation

• Encodes information in peripheral space for perfect retrieval
• Derives formulas for maximum noise-protected information capacity
• Establishes bounds on information scrambling time in dynamics